The category \(\text{MF}\left(\mathcal {X}(d), W\right)\)  is equivalent to the category of singularities \(D_{\text{sg}}\left(\mathcal {X}(d)_0\right)\)  of the zero fiber of \(\text{Tr}\,W\) .
The category \(\text{HA}(Q,W)\)  is monoidal with respect to \(m\) , see [1]}, and it is called the categorical Hall algebra (HA) of \((Q,W)\) . Its \(K_0\)  is called the K-theoretic Hall algebra (KHA) of \((Q,W)\) . For any edge \(e\in E\) , let \(\mathbb {C}^*\)  act on \(R(d)\)  by scaling the linear map corresponding to \(e\) . Let \(\left(\mathbb {C}^*\right)^E\)  be the product of these multiplicative groups.
One can consider a graded version of Hall algebras \(\text{HA}^{\text{gr}}(Q,W)\)  for any group \(\mathbb {C}^*\subset \left(\mathbb {C}^*\right)^E\)  scaling \(\text{Tr}\,W\) 
with weight 2.
